A162163 - OEIS

%I #9 Feb 24 2019 20:03:10

%S 179,467,739,809,1097,1171,1619,1801,1873,1907,2467,3203,3331,3491,

%T 3923,4051,4177,4211,4931,5507,5651,6067,6121,6353,6569,6659,7219,

%U 8081,8243,8297,8353,8819,9091,9161,9377,10243,10531,10657,10729,10889,11251,11699

%N Primes p such that p-1 and p+1 can individually be written as a sum of 2 and also as a sum of 3 distinct nonzero squares.

%C A subsequence of A162164.

%F {p=A000040(i): p-1 in A004431 and p-1 in A004432 and p+1 in A004431 and p+1 in A004432}. - _R. J. Mathar_, Jul 02 2009

%e p=12113: p-1=12112 = 36^2+40^2+96^2 = 36^2+104^2; p+1=12114 = 33^2+63^2+84^2 = 33^2+105^2.

%e p=4177: p-1=4176 = 24^2+60^2 = 24^2+36^2+48^2; p+1=4178 = 37^2+53^2 = 37^2+28^2+45^2. - _Vladimir Joseph Stephan Orlovsky_, Jun 26 2009

%e p=179: p-1=178 = 3^2+13^2 = 3^2+5^2+12^2; p+1=180 = 6^2+12^2=4^2+8^2+10^2. - _R. J. Mathar_, Jul 02 2009

%p isA004431 := proc(n) local x,y ; for x from 1 do if x^2 > n then RETURN(false); fi; y := n-x^2 ; if y> 0 and issqr(y ) then y := sqrt(y) ; if y <> x then RETURN(true) ; fi; fi; od: end:

%p isA004432 := proc(n) local x,y,z ; for x from 1 do if x^2 > n then RETURN(false); fi; for y from x+ 1 do if x^2+y^2>n then break ; fi; z := n-x^2-y^2 ; if z> 0 and issqr(z ) then z := sqrt(z) ; if z > y and z > x then RETURN(true) ; fi; fi; od: od: end:

%p for n from 1 to 2000 do p := ithprime(n) ; if isA004432(p-1) and isA004432(p+1) and isA004431(p-1) and isA004431(p+1) then printf("%d,",p) ; fi; od: # _R. J. Mathar_, Jul 02 2009

%t f[n_]:=Module[{k=1},While[(n-k^2)^(1/2)!=IntegerPart[(n-k^2)^(1/2)],k++; If[2*k^2>=n,k=0;Break[]]];k]; lst={};Do[p=Prime[n];x=p-1;y=p+1;If[f[x]> 0&&f[y]>0,a=x-(f[x])^2;b=y-(f[y])^2;If[f[a]>0&&f[b]>0,c=(x-(f[x])^2-(f[a])^2)^(1/ 2);d=(y-(f[y])^2-(f[b])^2)^(1/2);If[c!=f[x]&&c!=f[a]&&f[x]!=f[a], If[d!=f[y]&&d!=f[b]&&f[y]!=f[b],AppendTo[lst,p]]]]],{n,3,6*6!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Jun 26 2009 *)

%K nonn

%O 1,1

%A _Vladimir Joseph Stephan Orlovsky_, Jun 26 2009, Jun 27 2009

%E Definition corrected, Mathematica duplicate removed, missing values added by _R. J. Mathar_, Jul 02 2009